We develop a detailed and self-consistent numerical model for extremely-magnetised white dwarfs, which have been proposed as progenitors of overluminous Type Ia supernovae. This model can describe fully-consistent equilibria of magnetic stars in axial symmetry, with rotation, general-relativistic effects and realistic equations of state (including electron-ion interactions and taking into account Landau quantisation of electrons due to the magnetic field). We study the influence of each of these ingredients onto the white dwarf structure and, in particular, on their maximum mass. We perform an extensive stability analysis of such objects, with their highest surface magnetic fields reaching $\sim 10^{13}~G$ (at which point the star adopts a torus-like shape). We confirm previous speculations that although very massive strongly magnetised white dwarfs could potentially exist, the onset of electron captures and pycnonuclear reactions may severely limit their stability. Finally, the emission of gravitational waves by these objects is addressed, showing no possibility of detection by the currently planned space-based detector eLISA.